Interval Fuzzy Rule-Based Hand Gesture Recognition Benjam´ın R. Callejas Bedregal Depto de Inform´atica e Matem´atica Aplicada Universidade Federal do Rio Grande do Norte Campus Universit´ario, 59.072-970 Natal, Brazil
[email protected] Abstract Abstract. This paper introduces an interval fuzzy rulebased method for the recognition of hand gestures acquired from a data glove, with an application to the recognition of hand gestures of the Brazilian Sign Language. To deal with the uncertainties in the data provided by the data glove, an approach based on interval fuzzy logic is used. The method uses the set of angles of finger joints and of separation between finger for the classification of hand configurations, and classifications of segments of hand gestures for recognizing gestures. The segmentation of gestures is based on the concept of monotonic gesture segment, sequences of hand configurations in which the variations of the angles of the finger joints have the same sign (non-increasing or non-decreasing), separated by reference configurations that mark the inflexion points in the sequence. Each gesture is characterized by its list of monotonic segments. The set of all lists of segments of a given set of gestures determines a set of finite automata able to recognize such gestures.
1. Introduction Sign languages are the gestural languages used by deaf people in their daily face-to-face communication. Differently to the problems found in the processing of oral languages used by hearing people, the visual-gestural nature of sign languages gives rise to many specific problems for their automated recognition. There is an extensive literature about methods and systems for gesture recognition in general, and hand gesture recognition in particular, such as, e.g.: systems for the recognition of 3-D and 2-D gestures captured by different devices (data gloves, cameras etc.) [4] or systems for the graphical recognition of traces left on tablet devices [20]; methods based on fuzzy logic, neural networks or hybrid neuro-fuzzy methods [5]; fuzzy rule [24], finite state machine [13] or Hidden Markov Models [21] based methods.
Grac¸aliz P. Dimuro, Antˆonio C. Rocha Costa Programa de P´os-Graduac¸a˜ o em Inform´atica Universidade Cat´olica de Pelotas Felix da Cunha 412, 96010-000 Pelotas, Brazil {liz,rocha}@ucpel.tche.br
In particular, considering sign language recognition, some literature can be found related to fuzzy methods, such as, e.g, fuzzy decision trees [9] and neuro-fuzzy systems [1]. In this paper, we propose an interval fuzzy rule-based method for the recognition of hand gestures acquired from a data glove, extending the work presented in [2] to consider the uncertainties in the data provided by the glove. We apply the method to the recognition of hand gestures of LIBRAS, the Brazilian Sign Language [6]. The method uses the set of angles of finger joints and of the separation between fingers (given as intervals that enclose the uncertainties of the data obtained by the glove sensors) for the classification of hand configurations, and classifications of sequences of hand configurations for recognizing gestures. The segmentation of gestures is based on the concept of monotonic gesture segment, sequences of gestures in which the variations of the angles of the finger joints have the same sign (non-increasing or nondecreasing), separated by reference hand configurations that mark the inflexion points in the sequence. Each gesture is characterized by a list of monotonic segments, which determine a set of finite automata, which are able to recognize the gestures being considered. The paper is organized as follows. Section 2 presents a basic overview of fuzzy systems. Some concepts related to Interval Mathematics are discussed in Sect. 3. Our interval fuzzy rule-based method for hand gesture recognition is introduced in Sect. 4. The case study is presented in Sect. 5. The Conclusion is in Sect. 6.
2. Fuzzy Systems Fuzzy sets were introduced in 1965 by Zadeh [26] for representing vagueness in everyday life, providing an approximate and effective means for describing the characteristics of a system that is too complex or ill-defined to be described by precise mathematical statements. In a fuzzy approach the relationship between elements and sets follows a transition from membership to non membership that
is gradual rather than abrupt. Fuzzy set theory is the oldest and most widely used theory for soft computing, which deals with the design of flexible information processing systems[18], with applications in control systems [8], decision making [7], expert systems [23] etc. A fuzzy system implements a function (usually nonlinear) of n variables, given by a linguistic description of the relationship between those variables. Figure 1 illustrates the architecture of standard fuzzy systems. The fuzzificator computes the membership degrees of the crisp input values to the linguistic terms (fuzzy sets) associated to each input linguistic variable. The rule base contains the inference rules that associate linguistic terms of input linguistic variables to linguistic terms of output linguistic values. The information manager is responsible for searching in the rule base which rules are applicable for the current input. The inference machine determines the membership degrees of the output values in the output sets, by the application of the rules selected in the rule base. The defuzzificator gives a single output value as a function of the output values and their membership degrees to the output sets.
Fuzzificator
Rule Base
Defuzzificator
Information Manager Output Values
Input Values Inference Machine
Fuzzificator
Rule Base
Information Manager Input Values Inference Machine
Membership Degrees
Figure 2. A fuzzy rule based system Any real number x ∈ R that is uncertain for some reason (e.g., if it is obtained by a measuring instrument with limited resolution) is represented by a real interval X = [x1 ; x2 ], with x1 , x2 ∈ R and x1 ≤ x ≤ x2 . The set of intervals is denoted by IR. x1 and x2 denote, respectively, the left and right endpoints of X. A machine interval has floating point numbers as endpoints and outward roundings are used to guarantee that the resulting output interval of any computation process contains the actual result, with the range of the output interval being the indicative of the maximum error occurred in the process. The arithmetical operations ∗IR ∈ {+, −, ×, ÷} are defined, for all X, Y ∈ IR, as: X ∗IR Y = {x ∗ y | x ∈ X, y ∈ Y }.
(1)
For instance, if X = [x1 ; x2 ] and Y = [y1 ; y2 ], then X − Y = [x1 − y2 ; x2 − y1 ] [19].
Figure 1. A standard fuzzy systems However, many approximate methods do not produce a single final result, presenting several alternative solutions to a single problem (e.g., the different classes to which a given input may belong). Among them, several fuzzy rule-based methods for pattern recognition [18], fuzzy relations, fuzzy clustering, fuzzy neural systems etc. [17] etc. were developed, with applications to signature verification [12], and face recognition [16], for example. Such methods consider an architecture like the one shown in Fig. 2. Interval fuzzy rule-based systems consists of a generalization of such systems, by considering interval data type and interval membership degree values.
3. Interval Mathematics: some concepts Interval Mathematics [19] is a mathematical theory that aims at the automatic and rigorous controlling of the errors that arise in numerical computations.
The range of a real function f : R → R over a real interval X ∈ IR is given by f¯(X) = {f (x) | x ∈ X}.
(2)
An interval representation of f is an interval function F : IR → IR such that for each X ∈ IR, f (x) ∈ F (x) whenever x ∈ X [22]. Although there is not a unique interval representation for a given real function, it always hold that f¯(X) ⊆ F (X). It is not always possible to represent interval functions in the cartesian plan. However, in some cases, an interval function F (X) can be given by an interval of real functions F (X) = [inf{f (x) | x ∈ X}; sup{g(x) | x ∈ X}], (3) where f and g are real functions such that f ≤ g, which is denoted by [f, g] [22]. In this work, we consider that the membership functions are interval functions F expressed as in (3), with f = g, denoted by [f ], which is the best interval representation of
the function f , i.e., for any other interval representation F of f , [f ](X) ⊆ F (X), for any X ∈ R [22]. For the purpose of this work, the sign of a real interval X = [x1 ; x2 ] ∈ IR is defined as: + if x1 ≥ 0 and x2 > 0, − if x1 < 0 and x2 ≤ 0, sign([x1 , x2 ]) = (4) 0 otherwise. The absolute value of a real interval X = [x1 ; x2 ] ∈ IR is given as: | [x1 ; x2 ] |= max{| x1 |, | x2 |}.
(5)
For some applications of Interval Mathematics see [14].
4. The Interval Fuzzy Rule-Based Method
separation that characterizes a hand configuration. These angles are represented as interval angles [x − ²; x + ²] that enclose the uncertainties in the processing, where x is the angle given by a sensor and ² > 0, ² ∈ R, is the equipment tolerance, given by the manufacturer. In order to simulate this data transfer, a random generator of hand configurations was implemented, generating at each instant one hand configuration represented by a tuple of interval angles corresponding to each sensor (see Fig. 3): ((F1J1,F1J2,F1J3),S12,(F2J1,F2J2,F2J3),S23, (F3J1,F3J2,F3J3),S34,(F4J1,F4J2,F4J3),S45, (F5J1,F5J2,F5J3))
Given a hand configuration c and a sensor s, denote the interval value of each sensor angle by s(c), e.g., F1J1(c), S45(c) etc.
4.1 We consider a hypothetical data glove with 19 sensors, as shown in Fig. 3. The fingers are labelled as: F1 (little finger), F2 (ring finger), F3 (middle finger), F4 (index finger) and F5 (thumb). The joints in the fingers are labelled as J1 (the knuckle), J2 and J3, for each finger. A separation between two fingers is labelled as Sij to indicate that it is a separation between the fingers Fi and Fj. F3
F2
F1
J3
J3
J3
J3
J2
J2
J2
J2
J1
J1
J1
J1
F4
F5
S34
S23
S12
J3
J2 S45 J1
Figure 3. Localization of sensors in the data glove Since any movement can be represented as a sequence of frames, a hand movement using a data glove is represented as a sequence of hand configurations, one for each discrete time instant. That is, at each time instant, the data glove sensors should provide the set of angles of joints and finger
Fuzzification
Observe that when dealing with imprecise data represented by real intervals in IR, one has the problem that IR does not present a natural total order. Considering that fuzzy systems usually work with totally ordered data, the designer often has difficulty to express the membership degrees as a function in the cartesian plan, for interval-valued data. To solve this problem, interval fuzzy logic [3, 25] was introduced to deal with interval membership degrees, i.e., subintervals of [0;1] that allow the expression of the uncertainty of the expert about the classification of the data in linguistic terms. In this work, the membership functions are given as explained in Sect. 3, that is, interval functions of type [f ], where f is a real membership function. To each sensor corresponds a linguistic variable, whose values are linguistic terms representing typical angles of joints and separations. For the joints in the fingers (linguistic variables F1J1, F1J2, F1J3 etc.) the linguistic terms are: STRAIGHT (St), CURVED (Cv) and BENT (Bt). Figures 4 and 5 show the (interval) fuzzification for those variables. For the separations between fingers F1 and F2, F2 and F3, F4 and F5 (linguistic variable S12, S23, S45), the linguistic terms are: CLOSED (Cl), SEMI-OPEN (SOp) and OPEN (Op). For the separations between fingers F3 and F4 (linguistic variable S34), the linguistic terms are: CROSSED (Cr), CLOSED (Cl), SEMI-OPEN (SOp) and OPEN (Op). Figures 6 and 7 show the (interval) fuzzification for those variables.
4.2
The Interval Inference Process
The more generic and accepted way to consider fuzzy generalization of classical connectives are based on triangular norms (t-norms, t-conorms, fuzzy negations and fuzzy
STRAIGHT
CURVED
BENT
Membership Degree
1
0
-45
5
10
25
15 20
30
45 Angle (º)
Figure 4. Fuzzification of the linguistic variable of the joint F5J2 in the thumb finger F5 Membership Degree
STRAIGHT
0
7
BENT
CURVED
1
75 80
60
15
85 90 Angle (º)
Figure 5. Fuzzification of the linguistic variables of remaining finger joints CLOSED Membership Degree
SEMI-OPEN
OPEN
1
5
10
15
30
50
55
105 Angle (º)
0
Figure 6. Fuzzification of the linguistic variable of the separation S45 between the index finger F4 and the thumb finger F5 CLOSED
CROSSED
CLOSED
SEMI-OPEN
OPEN Membership Degree
Membership Degree
1
2
10
15
20
55 Angle (º)
0
5
-5 -1 1
SEMI-OPEN
1
2 5
10
15
20
OPEN
25
(a)
55 Angle (º)
0
(b) Figure 7. Fuzzification of the linguistic variables of the separations: (a) S34 between the middle finger F3 and the index finger F4, and (b) between remaining fingers
implications (residuum)) [15]. In [3, 25] those concepts were defined in an interval approach. In this paper, we use the generalization of the G¨odel tnorm, given by G([a, b], [c, d]) = [min{a, c}; max{b, d}].
(6)
For example, consider the (interval) membership function for the joints of the index finger F4 (Fig. 5) and the rule If F4J1 F4J2 F4J3 Then F4
is is is is
STRAIGHT and CURVED and CURVED StCvCv
If the angles provided by the data glove for the joints J1, J2 and J3 are 7o , 15o and 13o , respectively, and the tolerance is ² = 1o , then we have the following interval membership degrees: ¸ · 15 − 8 ϕF 4J1 ([6; 8]) = ; 1 = [0.875; 1], 8 · ¸ 14 − 7 ϕF 4J2 ([14; 16]) = ; 1 = [0.875; 1], 8 · ¸ 12 − 7 14 − 7 ϕF 4J3 ([12; 14]) = ; = [0.625; 0.875]. 8 8 Figure 8 illustrates the processes for obtaining the interval membership degree of joint J3 in the index finger F4. Then, using the interval t-norm G defined in (6), we obtain G(G([0.875; 1], [0.875; 1]), [0.625; 0.875]) = [0.625; 0.875] meaning that F 4 is in StCvCv with interval degree [0.625;0.875]. If one uses the product interval t-norm, i.e., P ([a; b], [c; d]) = [ac; bd],
(7)
the interval membership degree of finger F4 to StCvCv would be [0.546;0.875]. We observe that, in the fuzzification process, we considered only trapezoidal fuzzy sets and the interval G¨odel t-norm, motivated just by simplicity.
4.3
The Recognition Process
The hand gesture recognition process is divided into four steps: 1. Recognition of finger configurations; 2. Recognition of hand configurations;
3. Segmentation of the gesture in monotonic hand segments; 4. Recognition of the sequence of monotonic hand segments. For the Step 1 (recognition of finger configurations), 27 possible finger configurations are considered, for each finger. These configurations are codified in the format XYZ, where X, Y and Z are the values of the linguistic variables corresponding to the first joint J1, the second joint J2 and the third joint J3, respectively. For example, StStSt indicates that the three joints are STRAIGHT, StCvCv indicates that the first joint is STRAIGHT whereas the others are CURVED etc. The hand configuration is the main linguistic variable of the system, denoted by HC, whose linguistic terms are names of hand configurations, which names are application dependent. For instance, in Sect. 5, names of Brazilian Sign Language (LIBRAS) hand configurations (see Fig. 10) were used for such linguistic terms. The 27 possible finger configurations determine 27 inference rules that calculate membership degree of each finger to each configuration. For example, see the rule for the index finger in the previous subsection. Step 2 (recognition of hand configurations) determines the hand configuration, considering each finger configuration and separation between fingers. For example, the rule for the hand configuration [G] (Fig. 10) is: If F1 is BtBtSt F2 is BtBtSt F3 is BtBtSt F4 is StStSt F5 is StStSt Then HC is [G]
and and and and
S12 S23 S34 S45
is is is is
Cl Cl Cl Cl
and and and and
In Step 3 (segmentation of the gesture in monotonic hand segments), we divide each gesture in a sequence of k limit hand configurations l1 , . . . , lk , where l1 is the initial gesture configuration and lk is the terminal gesture configuration. The limit configurations are such that, for each sensor s and i = 1, . . . , k − 1, it holds that: (i) | s(li+1 ) − s(li ) |≤ 2², where the absolute value of an interval was defined in (5) and ² is the equipment tolerance. (ii) For each c between li and li+1 , sign(s(c) − s(li )) = sign(s(li+1 ) − s(li )). (iii) For each c0 after li+1 , sign(s(c0 ) − s(li+1 )) 6= sign(s(li+1 ) − s(li )),
MD
STRAIGHT
BENT
CURVED
[0.625;0.875]
1
7
15
75
60
80
85
90
0
Angle (º)
[12;14] Figure 8. Interval membership degree (MD) of joint J3 in the index finger F4 where the sign of an interval was given in (3). A sign equal to 0 is compatible with both negative and positive signs. The limit hand configurations are the points that divide the gesture into monotonic segments, that is, segments in which each sensor produces angle variations with constant (or null) sign. For each monotonic segment li li+1 , li and li+1 are its initial and terminal hand configurations, respectively. The procedure for step 3 is the following. To find any monotonic segment li li+1 , the next n configurations sent by the data glove after li are discarded, until a configuration cn+1 , such that sign(s(cn+1 ) − s(cn )) 6= sign(s(cn ) − s(l1 )) (or, cn+1 is the last configuration of the gesture). Then, cn (resp., cn+1 ) is the terminal hand configuration li+1 of the considered monotonic segment, and also coincides with the initial configuration of the next segment li+1 li+2 (if there is one). The process starts with li = l1 , which is the initial gesture configuration, and is repeated until the end of the gesture, generating the list of k limit hand configurations. In Step 4 (recognition of the sequence of monotonic hand segments), the recognition of each monotonic segment li li+1 is performed using a list of reference hand configurations r1 , r2 , . . . , rm that characterizes the segment, where r1 and rm are the initial and terminal hand configurations of the segment, respectively. A monotonic segment is recognized by checking that if it contains its list of reference hand configurations. The process is equivalent to a recognition based on a linear finite automaton (shown in Fig. 9), where li = r1 and li+1 = rm .
r1
r2
rm
Figure 9. Automaton for the recognition of monotonic segments
5. Case Study: Hand Gestures of LIBRAS As any other sign language, LIBRAS (L´ıngua Brasileira de Sinais – Brazilian Sign Language) is a natural language endowed with all the complexity normally found in the oralauditive languages. In the various works on automatic recognition of sign languages that have been developed along the years (see Sect. 1) the recognition of hand gestures has occupied a prominent place. To support that recognition process, a reference set of hand configurations is usually adopted, driven either from the linguistic literature on sign languages, or dynamically developed by the experimenters with an ad hoc purpose. For our purposes, we have chosen a standard set of hand configurations (some of them shown in Fig. 10), taken from the linguistic literature on LIBRAS [6]. Our method requires that each sign be thoroughly characterized in terms of its monotonic segments and the sequences of hand configurations that constitute such segments, and that the identification of the monotonic segments and hand configurations be manually provided to the system. Although a capture device such as a data glove can be used to help to identify the typical values of the angles of the finger joints, the final decision about the form of the membership functions that characterize the linguistic terms has to be explicitly taken and manually transferred to the
S1:
G1
G1X
X
CURIOUS
1
g
Figure 10. Some LIBRAS hand configurations [6]
system. We illustrate the application of the method by the definition of the necessary parameters for the recognition of hand gestures that constitute the signs CURIOUS and BIRD in LIBRAS. CURIOUS is a sign performed with a single hand placed right in front of the dominant eye of the signer, with the palm up and hand pointing forward. The initial hand configuration is the one named [G1] in Fig. 10. The gesture consists of the monotonic movement necessary to perform the transition from [G1] to [X] and back to [G1] again, such movements been repeated a few times (usually two or three). A possible analysis of the gestures that constitute the sign CURIOUS is: Initial configuration: [G1] Monotonic segment S1: [G1]-[G1X]-[X] Monotonic segment S2: [X]-[G1X]-[G1] State transition function for the recognition automaton: see Fig. 11. To support the recognition of the monotonic segments of CURIOUS, we have chosen to use one single intermediate hand configuration, [G1X], which does not belong to the
X
S2: S1
S2
S1
G1X
G1
S2
Figure 11. Automaton for the recognition of hand gestures of the sign CURIOUS.
reference set (Fig. 10) and whose characterization in terms of the set of membership functions for linguistic terms was defined in an ad hoc fashion, for the purpose of the recognition of CURIOUS. Together with [G1] and [X], it should be added to the list of hand configurations used by the recognition system. The sign BIRD is performed with one single hand, placed at the neutral signing space (mid arm distance from the signer), with the palm facing forward, fingers pointing up. A possible analysis of the hand gestures that constitute that sign is: Initial configuration: [bC] Monotonic segment S1: [bC]-[bO] Monotonic segment S2: [bO]-[bC] State transition function for the recognition automaton: see Fig. 12. Observe that, since the difference in angles of finger joints between [bC] and [bO] is small enough, it was not necessary to require from the recognition process the identification of any special intermediate hand configuration.
S1: BIRD
bC
bO S1
S2: S2
bO S1
bC S2
Figure 12. Automaton for the recognition of hand gestures of the sign BIRD.
6. Conclusion and Final Remarks This paper presented a fuzzy rule-based for the recognition of hand gestures. The method is highly dependent on a detailed previous analysis of the features of the gestures to be recognized, and on the manual transfer of the results of that analysis to the recognition system. This makes it suitable for the application to the recognition of hand gestures of sign languages, because of the extensive analysis that linguists that have already done of those languages.
Prototypes of a random gesture generator and of the gesture recognizer were implemented in the programming language Python, using the module PyInterval for Interval Mathematics [10, 11]. Future work is concerned with the recognition of arm gestures, by including the analysis of the angles of arm joints, so complete recognition of gestural features of signs can be achieved. Acknowledgments This work was partially supported by the Brazilian funding agency CNPq (Proc. 470871/2004-0, 470556/2004-8).
References [1] O. Al-Jarrah and A. Halawani. Recognition of gestures in arabic sign language using neuro-fuzzy systems. Artificial Intelligence, 133:117–138, 2001. [2] B. C. R. Bedregal, A. C. R. Costa, and G. P. Dimuro. Fuzzy rule-based hand gesture recognition. In M. Bramer, editor, Artificial Intelligence in Theory And Practice, number 217 in IFIP Series, pages 285–294, Boston, 2006. Springer. [3] B. C. R. Bedregal and A. Takahashi. The best interval representations of T-norms and automorphisms. Fuzzy Sets and Systems, 157(24):3220–3230, 2006. [4] O. Bimber. Continuous 6DOF gesture recognition: A fuzzylogic approach. In Proc. of the 7th Intl. Conf. in Central Europe on Computer Graphics, Visualization and Interactive Digital Media, volume 1, pages 24–30, 1999. [5] N. D. Binh and T. Ejima. Hand gesture recognition using fuzzy neural network. In Proc. of the ICGST Intl. Conf. on Graphics, Vision and Image Processing, pages 1–6, Cairo, 2005. [6] L. F. Brito. Por uma Gram´atica de L´ınguas de Sinais. Tempo Brasileiro, Rio de Janeiro, 1995. (in Portuguese). [7] C. Carlsson and R. Fuller. Fuzzy Reasoning in Decision Making and Optimization. Springer, Heidelberg, 2002. [8] G. Chen and T. T. Pham. Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems. CRC Press, Boca Raton, 2001. [9] G. Fang, W. Gao, and D. Zhao. Large vocabulary sign language recognition based on fuzzy decision trees. IEEE Transactions on Systems, Man and Cybernetics, 34:305– 314, 2004. [10] P. S. Grigoletti, G. P. Dimuro, and L. V. Barboza. An´alise intervalar de circuitos el´etricos. TEMA – Tendˆencias em Matem´atica Aplicada e Computacional, 7(2), 2006. (in Portuguese, to appear). [11] P. S. Grigoletti, G. P. Dimuro, and L. V. Barboza. M´odulo Python para matem´atica intervalar. In Proc. of XXIX CNMAC – Congresso Nacional de Matem´atica Aplicada e Computacional, Campinas, 2006. (in Portuguese, available at http://ppginf.ucpel.tche.br/gracaliz/papers). [12] M. Hanmandlu, M. H. M. Yusof, and V. K. Madasu. Offline signature verification and forgery detection using fuzzy modeling. Pattern Recognition, 38(3):341–356, 2005.
[13] P. Hong, M. Turk, and T. S. Huang. Gesture modeling and recognition using FSM. In Proc. of IEEE Conf. Face and Gesture Recognition, pages 410–415, Grenoble, 2000. [14] R. B. Keafort and V. Kreinovich, editors. Applications of Interval Computations. Kluwer, Boston, 1996. [15] E. P. Klement, R. Mesiar, and E. Pap. Triangular Norms. Kluwer Academic Press, Dordrecht, 2000. [16] K. Kwak and W. Pedrycz. Face recognition using a fuzzy fisherface classifier. Pattern Recognition, 38(10):1717– 1732, 2005. [17] C. T. Lin and C. S. G. Lee. Neural Fuzzy Systems: A neurofuzzy synergism to intelligent systems. Prentice Hall, Upper Saddle River, 1996. [18] S. Mitra and S. K. Pal. Fuzzy sets in pattern recognition and machine intelligence. Fuzzy Sets and Systems, 156:381–386, 2005. [19] R. E. Moore. Methods and Applications of Interval Analysis. SIAM, Philadelphia, 1979. [20] X. Ou, X. Chen and J. Yang. Gesture recognition for remote collaborative physical tasks using tablet PCs. In Proc. of the 9th IEEE Intl. Conf. on Computer Vision, Work. on Multimedia Tech. in E-Learning and Collaboration, Nice, 2003. [21] G. Rigoll, A. Kosmala, and S. Eickeler. High perfomance real-time gesture recognition using Hidden Markov Models. In I. Wachsmuth and M. Fr¨olich, editors, Gesture and Sign Language in Human-Computer Interaction, number 1371 in LNAI, pages 69–80, Berlin, 1998. Springer. [22] R. H. N. Santiago, B. C. Bedregal, and B. M. Aci´oly. Formal aspects of correctness and optimality of interval computations. Formal Aspects of Computing, 18:231–243, 2006. [23] W. Siler and J. J. Buckley. Fuzzy Expert Systems and Fuzzy Reasoning. John Wiley & Sons, New York, 2004. [24] M. Su. A fuzzy rule-based approach to spatio-temporal hand gesture recognition. IEEE Transactions on Systems, Man and Cybernetics, 30:276–281, 2000. [25] A. Takahashi and B. Bedregal. T-normas, T-conormas, complementos e implicac¸o˜ es intervalares. TEMA – Tendˆencias em Matem´atica Aplicada e Computacional, 7(1):139–148, 2006. (in Portuguese). [26] L. Zadeh. Theory of approximate reasoning. In J. Hayes, D. Michie, and L. I. Mikulich, editors, Machine Intelligence, pages 149–194. Ellis Horwood, 1970.
